Dynamical universality for random matrices

We establish an invariance principle corresponding to the universality of random matrices. More precisely, we prove the dynamical universality of random matrices in the sense that, if the random point fields $$ \mu ^N $$ μ N of N-particle systems describing the eigenvalues of random matrices or log-gases with general self-interaction potentials V converge to some random point field $$ \mu $$ μ , then the associated natural $$ \mu ^N $$ μ N -reversible diffusions represented by solutions of stochastic differential equations (SDEs) converge to some $$ \mu $$ μ -reversible diffusion given by the solution of an infinite-dimensional SDE (ISDE). Our results are general theorems that can be applied to various random point fields related to random matrices such as sine, Airy, Bessel, and Ginibre random point fields. In general, the representations of finite-dimensional SDEs describing N-particle systems are very complicated. Nevertheless, the limit ISDE has a simple and universal representation that depends on a class of random matrices appearing in the bulk, and at the soft- and at hard-edge positions. Thus, we prove that ISDEs such as the infinite-dimensional Dyson model and the Airy, Bessel, and Ginibre interacting Brownian motions are universal dynamical objects.